Paper Review

https://doi.org/10.3847/1538-4357/abfa9f

Summary

  • There may be a lag between X-ray and UV emission of AGNs but is hard to detect because of gaps in light curves.
  • GPs can interpolate the gaps and recover the lag.
  • Apply Structure Function Analysis to identify broken power law breakpoints.
  • Confirmed the lag is similar to previous studies
  1. Simulation study to select kernel
    • Matern \(\frac{1}{2}\) and Rational Quadratic
  2. Fit GP model to X-ray and UV light curves from Mrk 335
    • fit broken power law
    • estimate lag

Data

  • Narrow-line Seyfert 1 galaxy Mrk 335
  • Swift X-ray and UVW2 light curves
  • Time bins of one day
  • MJD 54327 to 58626
  • N = 509 (X-ray band)
  • N = 498 (UVW2 band)

Simulation Study

Identifying Flux distribution

Kolmogorov-Smirnov Test

\(H_0\): Sample was drawn from Gaussian distribution

At \(\alpha = 0.01\) significance level:

  • Fail to reject \(H_0\) for X-ray (p = 0.164).

  • Reject \(H_0\) for UV (p = \(1 \times 10^{-20}\)).

  • Fail to reject \(H_0\) for log transformed UV (p = 0.028)

OK, but perhaps unnatural to transform data to match likelihood.

Sim. Study Metholodology

  1. Simulate 1,000 light curves in X-ray and UV bands (4390 time points).
  2. Introduce unevenly spaced gaps in each light curve.
  3. Fit GP model with five different kernels (M 1/2, M 3/2, M 5/2, RQ, SE).
  4. Quantify fit using residual sum of squared errors (RSS)
  5. Calculate the negative log marginal likelihood (NLML)
  6. Average the RSS and NLML for each kernel.

Statistical Slip-up?

GP on Mrk 335 Observations

Notes about GP

  • Data is standardised before fitting, which effectively removes one of the kernel hyperparameters
  • Mean function set to zero
  • Kernels are stationary

Structure Function

PSD computation is complicated because of the uneven sampling of data so use structure function analysis instead.

\[\mathrm{SF}(\tau) = \frac{1}{N(\tau)} \sum_i \left[ f(t_i) - f(t_i + \tau)\right]^2\]

where \(f(t_i)\) is the count rate at \(t_i\), \(\tau = t_j - t_i\), and \(t_j \gt t_i\).

This is basically an analysis of autocorrelations.

Conclusions

  • GP kernels do not give same results therefore no strong evidence for break in X-ray power law.
  • Dips in structure function are due to uneven sampling.
  • Broad lag feature of up to 30 days.

Statistical Comments

Good

  • Very nice introduction to GPs.
  • Simulation study is well structured.

Statistical Comments

OK

  • Structure function analysis assumes stationarity.
  • Fitting power law to GP fit might be mispecified.
  • Uneven sampling eliminates PSD, but still causes problems for structure function.

Statistical Comments

Dubious

  • Gaussian likelihood for count rate data.
  • Correlation between RSS and NLML is a weak argument.
  • t-test for determining whether RSS was significant suffers from pseudo-replication and multiple-testing.

Bayes’ Theorem

\[ P(\theta | D) = \frac{P(D | \theta) \times P(\theta)}{P(D)}\]

\[\textrm{posterior} = \frac{\textrm{likelihood} \times \textrm{prior}}{\textrm{marginal likelihood}}\]

Bayesian Model Selection

The Bayes Factor, \(K\), is the ratio of two marginal likelihoods.

\[K = \frac{P(D \mid M_1)}{P(D \mid M_2)} = \dots = \frac{P(M_1 | D)}{P(M_2|D)} \cdot \frac{P(M_2)}{P(M_1)}\]

If the two models have the same prior probability, the Bayes Factor is equal to the ratio of the posterior probabilities.

If \(K \gt 1\) then \(M_1\) is more supported by the data than \(M_2\).

Negative Log Marginal Likelihood

\[\begin{align} - \log P(\boldsymbol{y}\mid\boldsymbol{t}, \theta) &= -\frac{1}{2}\boldsymbol{y}^\top (K_\theta(\boldsymbol{t},\boldsymbol{t}) + \sigma_y^2I)^{-1}\boldsymbol{y}\\ &-\frac{1}{2}\log | K_\theta(\boldsymbol{t},\boldsymbol{t}) + \sigma_y^2I|\\ &+ \frac{N}{2}\log(2\pi) \end{align}\]